Calc 2 Problem. Could someone help me understand the "u" notation in this integral?

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2 Answers
Apr 14, 2018

#int_1^0 12usqrt(u^2+1)du=-(8sqrt2-4)=4-8sqrt2#

Explanation:

The two provided integrals look almost exactly the same. In the first, the integrand is in terms of #x,# in the latter, the integrand has all #x# replaced with #u# and is otherwise the same.

The bounds have flipped between the two integrals. Fortunately, there is a rule for handling this:

#int_a^bf(x)dx=-int_b^af(x)dx#

Well, since we know #int_0^1 12xsqrt(x^2+1)dx=8sqrt2-4#, then, for the integral with flipped bounds and a similar integrand, we only have the same answer, but negative:

#int_1^0 12usqrt(u^2+1)du=-(8sqrt2-4)=4-8sqrt2#

Apr 14, 2018

The use of #u# is irrelevant.

Explanation:

The definite integral is a number. There is really no variable.

We are given

With #x# as the "dummy variable"

#int_0^1 12xsqrt(x^2+1) dx = 8sqrt2-4#

and also with #t# as the "dummy variable"

#int_0^1 12tsqrt(t^2+1) dt = 8sqrt2-4#

and with #u# as the "dummy variable"

#int_0^1 12usqrt(u^2+1) du = 8sqrt2-4#

and so on.

The question asks us to find

#int_1^0 12usqrt(u^2+1) du#

which is exactly the same as

#int_1^0 12xsqrt(x^2+1) dx#

and

#int_1^0 12tsqrt(t^2+1) dt#

and

#int_1^0 12rsqrt(r^2+1) dr#

As the answer by VNVDVI says, the integral is #-(8sqrt2-4) = 4-8sqrt2#.